In this paper, we establish some local universality results concerning the correlation functions of the zeroes of random polynomials with independent coefficients. More precisely, consider two random polynomials $f =\sum_{i=1}^n c_i ξ_i z^i$ and $\tilde f =\sum_{i=1}^n c_i \tilde ξ_i z^i$, where the $ξ_i$ and $\tilde ξ_i$ are iid random variables that match moments to second order, the coefficients $c_i$ are deterministic, and the degree parameter $n$ is large. Our results show, under some light conditions on the coefficients $c_i$ and the tails of $ξ_i, \tilde ξ_i$, that the correlation functions of the zeroes of $f$ and $\tilde f$ are approximately the same. As an application, we give some answers to the classical question `"How many zeroes of a random polynomials are real?" for several classes of random polynomial models. Our analysis relies on a general replacement principle, motivated by some recent work in random matrix theory. This principle enables one to compare the correlation functions of two random functions $f$ and $\tilde f$ if their log magnitudes $\log |f|, \log|\tilde f|$ are close in distribution, and if some non-concentration bounds are obeyed.

56 pages, no figures, to appear, IMRN. A gap in an argument invoking Gromov type theorems (in the treatment of the Kac model) has been fixed